GEOMETRIC ANALYSIS OF AN OBSERVER ON A SPHERICAL EARTH AND AN AIRCRAFT OR SATELLITE Michael Geyer S π/2 - α - θ d h α N U Re Re θ O Federico Rostagno Michael Geyer Peter Mercator nosco.com Project Memorandum — September 2013 DOT-VNTSC-FAA-13-08 Prepared for: Federal Aviation Administration Wake Turbulence Research Program DOT/RITA Volpe Center TABLE OF CONTENTS 1. INTRODUCTION...................................................................................................... 1 1.1 Basic Problem and Solution Approach........................................................................................1 1.2 Vertical Plane Formulation ..........................................................................................................2 1.3 Spherical Surface Formulation ....................................................................................................3 1.4 Limitations and Applicability of Analysis...................................................................................4 1.5 Recommended Approach to Finding a Solution.........................................................................5 1.6 Outline of this Document ..............................................................................................................6 2. MATHEMATICS AND PHYSICS BASICS ............................................................... 8 2.1 Exact and Approximate Solutions to Common Equations ........................................................8 2.1.1 The Law of Sines for Plane Triangles.........................................................................................8 2.1.2 The Law of Cosines for Plane Triangles.....................................................................................8 2.1.3 Solution of a Quadratic Equation ................................................................................................9 2.1.4 Computing Inverse Trigonometric Functions .............................................................................9 2.1.5 Expansions of arcsin, arccos and arctan for Small Angles ....................................................... 11 2.1.6 Surface Area on a Sphere ..........................................................................................................11 2.2 Shape of the Earth ....................................................................................................................... 11 2.2.1 WGS-84 Ellipsoid Parameters................................................................................................... 11 2.2.2 Radii of Curvature Defined in Literature .................................................................................. 12 2.2.3 Vincenty’s Algorithm for the Direct and Indirect Problems of Geodesy..................................14 2.3 Altitude-Related Matters ............................................................................................................ 14 2.3.1 Accounting for User Altitude....................................................................................................15 2.3.2 Conditions for Unblocked US LOS...........................................................................................15 2.3.3 Different Notions of Altitude ....................................................................................................16 3. SOLUTION TO VERTICAL PLANE FORMULATION............................................ 18 3.1 Mathematical Problem and Solution Taxonomy......................................................................18 3.1.1 Mathematical Formulation ........................................................................................................ 18 3.1.2 Taxonomy of Solution Approaches...........................................................................................18 3.1.3 Detailed Geometry.....................................................................................................................19 3.2 Computing Geocentric Angle .....................................................................................................20 3.2.1 Altitude and Elevation Angle Known – Basic Method............................................................. 20 3.2.2 Altitude and Elevation Angle Known – Alternative Method....................................................21 3.2.3 Altitude and Slant Range Known.............................................................................................. 21 3.2.4 Elevation Angle and Slant Range Known................................................................................. 22 3.3 Computing Elevation Angle........................................................................................................ 22 3.3.1 Altitude and Geocentric Angle Known.....................................................................................22 3.3.2 Altitude and Slant Range Known.............................................................................................. 23 3.3.3 Geocentric Angle and Slant Range Known...............................................................................23 3.4 Computing Slant Range .............................................................................................................. 23 3.4.1 Altitude and Geocentric Angle Known.....................................................................................23 3.4.2 Altitude and Elevation Angle Known ....................................................................................... 24 3.4.3 Geocentric Angle and Elevation Angle Known ........................................................................ 24 i DOT/RITA Volpe Center 3.5 Computing Altitude .....................................................................................................................24 3.5.1 Slant Range and Geocentric Angle Known...............................................................................24 3.5.2 Slant Range and Elevation Angle Known................................................................................. 25 3.5.3 Elevation Angle and Geocentric Angle Known ........................................................................ 25 3.6 Example Applications.................................................................................................................. 25 3.6.1 Example 1: En Route Radar Coverage.....................................................................................26 3.6.2 Example 2: Aircraft Instrument Approach Procedure.............................................................. 29 3.6.3 Example 3: Satellite Visibility of/from Earth ..........................................................................31 4. SOLUTION TO SPHERICAL SURFACE FORMULATION.................................... 33 4.1 Basics of Spherical Trigonometry.............................................................................................. 33 4.1.1 Historical Perspective................................................................................................................33 4.1.2 Application to Navigation ......................................................................................................... 33 4.1.3 Resources Available on the Web...............................................................................................34 4.1.4 Key Formulas ............................................................................................................................ 35 4.1.5 Taxonomy of Problems Involving Spherical Triangles............................................................. 37 4.1.6 Taxonomy of Spherical Surface Formulation Problems ...........................................................38 4.2 Solving the Indirect Problem of Geodesy.................................................................................. 39 4.2.1 Computing the Geocentric Angle.............................................................................................. 39 4.2.2 Computing the Azimuth Angles of the Connecting Arc ...........................................................40 4.3 Solving the Direct Problem of Geodesy .....................................................................................41 4.3.1 Computing the Satellite Latitude...............................................................................................41 4.3.2 Computing the Satellite Longitude............................................................................................ 42 4.3.3 Computing the Azimuth of the Connecting Arc at the Satellite................................................42 4.4 Solving a Modified Version of the Direct Problem: Satellite Longitude Known .................. 43 4.4.1 Computing the Satellite Latitude...............................................................................................43 4.4.2 Computing the Geocentric Angle.............................................................................................. 43 4.4.3 Computing the Azimuth of the Connecting Arc at the Satellite................................................44 4.5 Solving a Modified Version of the Direct Problem: Satellite Latitude Known .....................44 4.5.1 Computing the Azimuth of the Connecting Arc at the Satellite................................................44 4.5.2 Computing the Satellite Longitude............................................................................................ 45 4.5.3 Computing the Geocentric Angle.............................................................................................